Static and dynamic magnetic properties of spherical magnetite

JOURNAL OF APPLIED PHYSICS

VOLUME 94, NUMBER 5

1 SEPTEMBER 2003

Static and dynamic magnetic properties of spherical magnetite nanoparticles G. F. Goya,a) T. S. Berquo´, and F. C. Fonseca ˜ o Paulo, CP 66318, 05315-970, Sa ˜ o Paulo, SP, Brazil Instituto de Fı´sica, Universidade de Sa

M. P. Morales Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049-Madrid, Spain

共Received 28 March 2003; accepted 23 June 2003兲 We present a detailed study of static and dynamic magnetic behavior of Fe3 O4 nanoparticles with average particle sizes 具d典 ranging from 5 to 150 nm. Bulk-like properties such as saturation magnetization, hyperfine parameters, coercive field, and Verwey transition are observed in 150 nm particles. For decreasing particle size, the Verwey temperature, T V , shifts down to ⬃20 K for 具 d 典 ⫽50 nm and is no longer observable for smaller particles. The smallest particles ( 具 d 典 ⫽5 nm) display superparamagnetic behavior at room temperature, with transition to a blocked state at T B ⬃45 K, which depends on the applied field. The existence of surface spin disorder can be inferred from the decrease of saturation magnetization M S at low temperatures, as the average particle size is reduced. This disordered surface did not show effects of exchange coupling to the particle core, as observed from hysteresis loops after field cooling in a 7 T magnetic field. For particles with 具 d 典 ⫽5 nm, dynamic ac susceptibility measurements show a thermally activated Arrhenius–Neél dependence of the blocking temperature with applied frequency. The interparticle interactions are found to influence the energy barriers yielding an enhancement of the estimated magnetic anisotropy. From the calculus of the magnetic anisotropy, it is inferred that there is no structural transition from cubic to triclinic symmetry for 具 d 典 ⫽5 nm, in agreement with the absence of the Verwey transition. A value K 1 ⫽4.68⫻105 erg/cm3 is obtained for the magnetocrystalline anisotropy constant of the cubic phase. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1599959兴

I. INTRODUCTION

erties of great potential for applications in giant magnetoelectronic and spin-valve devices based on magnetite films. In spite of the fact that magnetite is perhaps the oldest magnetic material known, some aspects on the basic mechanisms related to the Verwey transition are still being discussed, especially in nanosized systems.11,12 An additional issue to be solved for potential applications is that the magnetic properties of magnetite-based nanostructured systems 共particles or films兲 strongly depend on the synthesis route,13–15 as well as on the nonmagnetic matrix 共or substrate兲 chosen. Moreover, for a given synthesis technique the final magnetic properties seem to depend heavily on tiny changes in the local structure like antiphase boundaries 共APB兲, oxygen deficiency, and local ionic disorder.16 –18 Above the Verwey temperature, T V ⬃120 K, magnetite has a cubic spinel structure 共space group Fd3m), with lattice parameter a 0 ⬃8.397 Å and the O atoms arranged in a facecentered-cubic 共fcc兲 lattice. The lattice can accommodate Fe3⫹ on the tetrahedral site 共A兲 and Fe3⫹ and Fe2⫹ on the octahedral site 共B兲 in antiparallel arrangement, yielding ferrimagnetic order below T C . Bulk magnetite has cubic magnetic anisotropy, with the 具1 1 1典 and 具1 0 0典 directions being the easy and hard axes of magnetization, respectively. At room temperature 共RT兲, the first-order magnetocrystalline anisotropy constant has a negative value (K 1 ⫽⫺1.35⫻105 erg/cm3 ) that changes sign at low temperature, passing through an isotropic point at a temperature few degrees

Nanosized magnetic structures are currently key materials for advancements in electronics,1 optoelectronics,2 magnetic storage,3 and many bioinspired applications.4 – 6 What is usually termed ‘‘nanostructured systems’’ comprises those materials whose properties are determined by entities 共particles, crystallites, or clusters兲 with characteristic lengths between 1 and 100 nm in at least two dimensions. If the grain or domain size becomes comparable or smaller to the characteristic length scale of the interaction processes controlling a particular property, different effects and unusual chemical and physical properties can be expected that are highly attractive in a number of technical applications.7–9 In recent times, large advancements have been achieved related to the synthesis and characterization of well-defined, discrete magnetic nanoparticles for both fundamental and technological purposes. However, precise knowledge of the relationships between particle shape and size distribution, surface structure, and the resulting magnetic properties of magnetic nanoparticles is still lacking. This remains true even for particles composed of ‘‘simple’’ pure materials such as Fe, Co, or Ni, whose bulk properties are well understood.10 Magnetite Fe3 O4 has recently attracted attention because bulk Fe3 O4 has a high Curie temperature (T C ⬃850 K) and nearly full spin polarization at room temperature, both propa兲

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J. Appl. Phys., Vol. 94, No. 5, 1 September 2003

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above the Verwey transition.19 On cooling below T V , the change from cubic to triclinic structure yields a change to uniaxial anisotropy with 具0 0 1典 easy axis.20 Since the magnetic hardness of single-domain particles results from the concurrent effects of shape and crystal anisotropies, there is an obvious need for separating these effects to study each of them. As the magnetocrystalline anisotropy is associated with each material and cannot be eliminated, we have chosen a simple way for disentangling these effects by making spherical particles to eliminate the shape anisotropy. In this work, we present a detailed structural and magnetic characterization of spherical Fe3 O4 nanoparticles with different mean sizes from 5 to 150 nm, aiming to study the relation between structural and magnetic properties in the blocked state as well as in the superparamagnetic 共SPM兲 regime. II. EXPERIMENTAL PROCEDURE

Four magnetite (Fe3 O4 ) samples have been prepared by different routes and named according to their mean particle size 共in nanometers兲: samples M150, M50, M10, and M5. Sample M150 was prepared by first precipitating hematite particles by forced hydrolysis of a 0.02 M FeCl3 solution at 100 °C during 2 days21 and further reduction under 1 atm of hydrogen at 360 °C during 3.5 h. Sample M50 was obtained after heating at 90 °C a solution of 0.1 M KOH and 0.05 M FeSO4 during 24 h, in the presence of 0.2 M of KNO3 . 22 Sample M50 was washed with HCl and later with water. Sample M10 was prepared by adding 50 ml of an aqueous solution 共0.33 M FeCl2 and 0.66 M FeCl3 ) to 450 ml of 1 M ammonium hydroxide under constant stirring at room temperature.23 During preparation of both M10 and M50 samples, N2 gas was flowed previously through the base solution and the formed black precipitate was washed and dried at 40 °C during 2 h in air. Sample M10 was washed with water. Smaller particles of around 5 nm were obtained following the Lee et al. method.24 The iron salt mixture, Fe共II兲 and Fe共III兲, was added to 1 M K共OH兲 solution with 1 wt % of polyvinylalcohol 共PVA兲 at room temperature. X-ray diffraction 共XRD兲 measurements were performed using a Philips 1710 powder diffractometer using Cu K ␣ radiation in the 2␪ range from 5° to 70°. Transmission electron microscopy 共TEM兲 images were taken in a highresolution 200 keV JEOL-2000 FXII microscope, in order to analyze the structure and morphology of the magnetic powders. Mo¨ssbauer spectroscopy 共MS兲 measurements were performed with a conventional constant-acceleration spectrometer in transmission geometry with a source of about 50 mCi 57 Co in a Rh matrix between 78 and 296 K. Hyperfine parameters such as the distribution of hyperfine magnetic field, isomer shift, and quadrupole shift have been determined by the NORMOS program, and ␣-Fe at 296 K was used to calibrate isomer shifts and velocity scale. Room temperature M vs H cycles were performed in a vibrating sample magnetometer using an electromagnet to produce fields up to 2 T. Static and dynamic magnetic measurements as a function of frequency and temperature were performed in a commercial superconducting quantum interference device 共SQUID兲 mag-

FIG. 1. TEM pictures for samples M5 共a兲, M10共b兲, M50 共c兲, and M150 共d兲.

netometer 共Quantum Design兲. Zero-field-cooled 共ZFC兲 and field-cooled 共FC兲 curves were taken between 5 and 400 K, for different values of cooling field H FC (10 Oe⬍H ⬍70 kOe). Data were obtained by first cooling the sample from room temperature in zero applied field 共ZFC process兲 to the basal temperature 共5 K兲. Then, a field was applied and the variation of magnetization was measured with increasing temperature up to T⫽400 K. After the last point was measured, the sample was cooled again to the basal temperature keeping the same field 共FC process兲; then, the M vs T data were measured for increasing temperatures. III. EXPERIMENTAL RESULTS

Analysis of TEM images 共Fig. 1兲 showed that the resulting samples consist of homogeneous particles with nearly spherical shape, increasing the size from M5 to M150 samples. The resulting average size estimated from these images are displayed in Table I. The x-ray diffractograms for the four samples shown in Fig. 2 could be indexed with the Fe3 O4 single phase, except for the M50 sample where small peaks corresponding to the main reflections of goethite were observed. It can be noticed that the lines broaden steadily from the M150 to M5 sample as a result of the decrease in the average particle size, in agreement with TEM data. We have estimated the average grain size 具d典 from the main reflections of each diffractogram, by using the Scherrer formula and without considering possible contributions of crystal stress to the observed linewidth. It can be seen from Table I that the agreement between TEM and XRD data is very good, giving support to the following discussion on the magnetic properties of these particles. A. Mo¨ssbauer study

The Mo¨ssbauer spectra of Fe3 O4 particles with different sizes, at 296 and 78 K, are shown in Figs. 3 and 4, respectively. A first, general feature that can be observed at RT is


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TABLE I. Average particle diameters 具d典 extracted from TEM and x-ray data, saturation magnetization M S , coercive field H C , remanence M R , and the ratio R⫽M R /M S for Fe3 O4 samples measured at 5 and 300 K. Particle Crystal size size 共TEM兲共x-ray兲 Temp Sample 共nm兲共nm兲共K兲

HC 共Oe兲

MS MR 共emu/g兲共emu/g兲 M R /M S

M5

4

4.3

5 300

294共10兲 12共7兲

56.1共2兲 31.8共2兲

10.1共8兲 0

0.15 0

M10

11.5

⬃10

5 300

202共10兲 34共7兲

77.5共2兲 60.1共2兲

12.7共8兲 3.9共2兲

0.16 0.016

M50

47.7

55

5 300

211共10兲 156共7兲

77.8共2兲 65.4共2兲

16.3共8兲 16.4共2兲

0.21 0.25

M150

⬃150

⬃150

5 300

736共10兲 323共7兲

88.5共2兲 75.6共2兲

32.1共8兲 18.9共2兲

0.36 0.25

the gradual passage from bulk-like to SPM behavior, as the average particle size decreases. For sample M150, the room temperature spectrum was fitted using two magnetic components of hyperfine fields B hyp⫽49.5 and 46.0 T, corresponding to Fe3⫹ ions at sites A and (Fe2⫹ Fe3⫹ ) ions at site B, respectively 共see Table II兲, with nearly null quadrupole splitting 共QS兲 and intensity ratio B:A⬃1:1. These well-known features of MS are related to the electron transfer process 共electron hopping兲 between Fe2⫹ and Fe3⫹ ions on the octahedral B site taking place for T⬎115– 120 K 共Verwey transition兲. The absence of SPM signal 共doublet兲 in the MS shows that these particles have a multidomain structure, in agreement with the theoretical estimation for maximum linear dimensions of single-domain particles of ⬃128 nm.25,26 As particle size decreases, both hyperfine fields merge into a single sextet 共sample M50兲, and for sample M10 the thermal relaxation effects are clearly observed 共Fig. 3兲. Ad-

FIG. 2. X-ray diffraction patterns of the Fe3 O4 samples with different average particle sizes.

FIG. 3. Mo¨ssbauer spectra 共open circles兲 recorded at 296 K for the Fe3 O4 samples. The solid lines are the best fit to experimental data, and dotted lines represent each component of the total fit.

ditionally, sample M50 showed a small component from goethite (B hyp⫽37.7 T), amounting to a ⬃3% of the total spectral area, in agreement with XRD data. Sample M5 displays only a central doublet corresponding to a full superparamagnetic state.

FIG. 4. Left panel: Mo¨ssbauer spectra 共open circles兲 recorded at 78 K for the Fe3 O4 samples. Solid lines are the best fits using hyperfine field distributions P(B hyp) shown on the right panel.


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TABLE II. Hyperfine parameters extracted from Mo¨ssbauer spectra at T ⫽296 K: hyperfine field (B hyp), quadrupolar splitting 共QS兲, isomer shift 共IS兲, and spectral area 共I兲 for samples M5, M10, M50, and M150. T 共K兲

Sample

296

M5 M10 M50 M150

Site

B hyp 共T兲

QS 共mm/s兲

IS 共mm/s兲

I 共%兲

A B

46.0 50.0 49.5 46.0

0.60共1兲 0.00共1兲 0.00共1兲 0.05共1兲 0.02共1兲

0.22共1兲 0.21共1兲 0.22共1兲 0.20共1兲 0.64共1兲

100共3兲 100共3兲 100共3兲 57共3兲 43共3兲

At T⫽78 K 共i.e., below the Verwey transition兲 the spectrum of sample M150 shows the change of lattice symmetry to the low-temperature 共triclinic兲 phase.20,27 We attempted to fit the spectrum with two distributions of hyperfine fields, corresponding to Fe3⫹ ions at A sites and to Fe2⫹ and Fe3⫹ at B sites. The resulting distributions are shown on the right side of Fig. 4. There is not general agreement about the type of fit that best reflects the properties of the magnetite phase at low temperatures, and different authors have proposed from two to more than five magnetic components for T ⬍T V . 28,29 However, there are several common features observed in almost all low-T spectra reported, such as the increase of B hyp and the nearly zero QS values. For samples M50 and M10, the spectra at 78 K showed that the thermal fluctuations observed at room temperature are no longer present. Moreover, using a second site distribution did not improve the fits, therefore, these spectra were fitted using a single magnetic sextet. The resulting distribution profiles showed a maximum centered at B hyp⬃52.3 and 51.2 T for M50 and M10, respectively. For sample M5 thermal relaxation effects are still dominant at this temperature, in agreement with the blocking temperature of ⬃45 K observed from magnetization data 共see below兲. B. Magnetization study

To better understand the magnetic properties of these systems we performed magnetization measurements as a function of temperature and applied fields. The ZFC/FC curves 共Fig. 5兲 of the M150 sample show clearly that the Verwey transition is still present, although at a lower temperature (T V ⫽98(1) K). Sample M50 still shows a kink at ⬃16 K, probably related to the Verwey transition, which is shifted to a lower temperature due to size effects. Additionally, a broad maximum is observed in the ZFC sample centered at T B ⬃300 K, suggesting a blocking process of large particles. For M10 and M5 samples the maximum at T B shifts to 107共1兲 and 45共1兲 K, respectively, and the Verwey transition is no longer observable. The dependence of the high-field magnetization with temperature is shown in Fig. 6. Since the magnetization of the sample at 70 kOe is only ⬃5% below the saturation value M S extracted from extrapolation to infinite field, we can consider that M S (T)⫽M (T,H⫽70 kOe). It can be seen from Fig. 6 that M S (T) steadily decreases with increasing T, and this thermal decrease could be fitted by the Bloch law M S 共 T 兲 ⫽M S 共 0 兲关 1⫺B 0 T 3/2兴 ,

FIG. 5. Zero-field-cooled 共ZFC兲 and field-cooled 共FC兲 curves for samples of different particle sizes, taken with H FC⫽500 G.

where M S (0) is the saturation magnetization at zero temperature. This is in contrast to previous results reported on a system of magnetite particles suspended in a ferrofluid,30 where a T 2 dependence 共slightly modified by a T-dependent prefactor兲 was proposed. The inset of Fig. 6 clearly shows that the linear relation between 关 M S ⫺M S (T) 兴 /M S vs T 3/2 is followed along the whole experimental temperature range (5 K⭐T⭐350 K). Although theoretical calculations31 have predicted a thermal dependence of the form T ␣ with 3/2 ⬍ ␣ ⬍3 for small ferromagnetic clusters, many nanostructured systems such as ␣-Fe particles in the SiO2 matrix32 and ␥ -Fe2 O3 nanoparticles33 have provided experimental evidence for the T 3/2 dependence at least up to room temperature. For the present magnetite nanoparticles, the prefactor B 0 ⫽3.3⫻10⫺5 K⫺3/2 obtained from the fitting procedure is

FIG. 6. Magnetization vs temperature for M5 sample taken at H⫽70 kOe 共open circles兲. The solid line is the fit using the Bloch law. Inset: graph M (T) vs T 3/2 to show the linear relationship up to ⬃350 K.


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FIG. 7. Magnetization hysteresis curves measured at 296 K for the Fe3 O4 samples.

similar to the 2.8⫻10⫺5 K⫺3/2 observed for ␥ -Fe2 O3 particles.33 Furthermore, the similarity of B 0 values in Fe3 O4 and ␥ -Fe2 O3 nanoparticles can be understood from 共a兲 their common cubic crystal structure 共space group O7h -Fd3m); 共b兲 their close lattice parameters a 共⬃8.35– 8.39 Å兲; and 共c兲 their similar ordering temperatures 共⬃600 °C兲 that involve similar exchange integral values J.34 As pointed out by Martıńez et al.,33 the T 3/2 dependence indicates the existence of spinwave excitations in these nanometric domains of fewnanometer size. Using a very simplified landscape for the thermal excitation of the magnons at low temperatures, the prefactor B of the Bloch law can be expressed as B ⫺1 0 ⬃D ⫽2a 2 JS, where D is the spin-wave stiffness constant, and S is the spin of magnetic ions. From this relationship we extracted an average value of J/k B ⫽23 K, which is of the same order of magnitude that the exchange constant J AB between A and B sublattices in bulk magnetite.34 Hysteresis loops performed at room temperature 共Fig. 7兲 show that the coercivity decreases with decreasing particle size, attaining zero value for the M5 sample, in agreement with the SPM behavior for a 具 d 典 ⫽5 nm particle size. It is also observed that samples M150 and M50 are already saturated at H⫽20 kOe, whereas samples M10 and M5 show still slightly increasing magnetic moment. For the latter two samples, the M S values 共Table I兲 were obtained from the relationship M ⫽M S (1⫺ ␤ /H), where ␤ is a fieldindependent parameter, after extrapolating to infinite field. The results obtained differ from the M (H⫽20 kOe) values by 3% and 11% for the M10 and M5 samples, respectively. In the SPM regime, the magnetization M sp(H) plotted against H/T results in a universal curve, described by the relation M sp⫽N ␮ L(x), where N is the number of particles of magnetic moment ␮, and L(x) is the Langevin function of argument x⫽ ␮ H/kT, i.e., the ratio of magnetic to thermal energy. To take into account the effects of size dispersion always present in any real system, the magnetization of SPM grains in a magnetic field H is better described as a weighted sum of Langevin functions M sp⫽M bulk S

冕冉␮ 冊 ⬁

L

0

H f 共 ␮ 兲d␮, kT

共1兲

FIG. 8. Superparamagnetic contribution M sp of sample M5 at T⫽296 K. The solid line is the fitted curve obtained from Eqs. 共1兲 and 共2兲. Inset: resulting distribution function with D 0 ⫽5.0 nm and ␴ ⫽1.4.

where f ( ␮ ) is a log–normal distribution 1

f 共 ␮ 兲⫽

冑2 ␲ ␴ ␮

冉

exp ⫺

共 ln ␮ / ␮ 0 兲 2 2␴2

冊

共2兲

of magnetic moments ␮. In Eq. 共2兲, ␴ is the distribution width and ␮ 0 is the median of the distribution related to the mean magnetic moment ␮ m by ␮ m ⫽ ␮ 0 exp(␴2/2). The distribution function satisfies the normalization condition M S⫽

冕

⬁

0

f 共 ␮ 兲d␮.

In order to obtain the magnetic moment distribution for the M5 sample, we have fitted the M (H) data using Eqs. 共1兲 and 共2兲, which contains two free parameters ( ␮ 0 and ␴兲. The result is shown in Fig. 8. To relate the moment distribution to the particle size distribution we have used the saturation magnetization value M S ⫽398.7 emu/cm3 , as extracted from Table I. Also, based on the TEM data, we have presumed spherical geometry for particles, i.e., V⫽ ␲ D 3 /6 where D is the particle diameter. With these assumptions we obtained the values ␮ 0 ⫽2805 ␮ B and D 0 ⫽5.0 nm, the latter being in excellent agreement with estimations from TEM and x-ray diffraction. Fitting the M (H) data with a simple Langevin function yielded values ␮ ⫽6599 ␮ B and D⫽6.7 nm, which are slightly larger than the values from the log-normal distribution f ( ␮ ) of magnetic moments. The different values reflect mainly the difference between median ( ␮ 0 ) and mean ( ␮ m ) values of the magnetic moment distribution, related to the distribution width ␴ 2 ⫽2 ln(␮m /␮0). In order to see possible effects of exchange coupling between core and surface of the particles, hysteresis loops were also measured at 5 K after field cooled in a 7 T field. As shown in Fig. 9, the cycles are symmetric within experimental error 共i.e., no exchange bias can be observed兲, indicating the lack of exchange coupling between particle surface and core. At low temperatures, it is observed that the saturation magnetization decreases for smaller particles, indicating increasing spin disorder effects at the particle surface, yielding a smaller net magnetic moment.


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FIG. 9. Magnetization hysteresis curves measured at 5 K for the Fe3 O4 samples in field-cooling mode with H FC⫽7 T. Inset: amplification of the low-field region showing the symmetry of the loops.

C. Dynamic properties

Turning now to the dynamics of the magnetic particle systems, Fig. 10 displays the temperature dependence of ␹ ⬘ (T) and ␹ ⬙ (T) of the smallest particles 共sample M5兲 for different frequencies f ranging from 20 mHz to 1 kHz. The data for both components ␹ ⬘ (T) and ␹ ⬙ (T) exhibit the expected behavior of SPM systems, i.e., the occurrence of a maximum at a temperature T m for both ␹ ⬘ (T) and ␹ ⬙ (T) components, which shifts towards higher values with increasing frequency.35 As a useful criterion for classifying the freezing/blocking process observed, we used the empirical parameter ⌽, which represents the relative shift of the temperature T m per decade of frequency. For the M5 sample, we obtain ⌽⫽

⌬T m ⫽0.07, T m ⌬ log10共 f 兲

where ⌬T m is the difference between T m measured in the ⌬ log10( f ) frequency interval. This value is close to the 0.10 value found for SPM systems, providing a modelindependent classification of the thermally activated origin of

FIG. 10. Temperature dependence of the in-phase 共real兲 component ␹ ⬘ (T) of the magnetic susceptibility for the M5 sample, at different excitation frequencies. Arrows indicate increasing frequencies. Inset: Out of phase 共imaginary兲 component ␹ ⬙ (T). The data were taken with an external magnetic field H⫽5 Oe.

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FIG. 11. Arrhenius plot of the relaxation time ␶ vs inverse blocking temperature T ⫺1 B obtained from the imaginary component ␹ ⬙ (T) as extracted from Fig. 9. Dotted line is the best fit using Eq. 共3兲 with ␶ 0 ⫽9⫻10⫺13 s and E a ⫽1690 K.

the observed transition. On the other side, it is known that smaller values of ⌽ usually result from spin-glass-like surface behavior or interparticle interactions.35–37 The latter corresponds more likely to the present case, since our samples are nondiluted particulate systems. The dynamic response of an ensemble of fine particles is determined by the measuring time ␶ m 共or frequency兲 of each experimental technique. As the reversion of the magnetic moment in a single-domain particle over the anisotropy energy barrier E a is assisted by thermal phonons, the relaxation time ␶ exhibits an exponential dependence on temperature characterized by a Neél–Arrhenius law

␶ ⫽ ␶ 0 exp

冉冊

Ea , k BT

共3兲

where ␶ 0 is in the 10⫺9 – 10⫺11 s range for SPM systems. In the absence of an external magnetic field, the energy barrier can be assumed to be proportional to the particle volume V, E a ⫽K effV sin2 ␪, where K eff is an effective magnetic anisotropy constant and ␪ is the angle between the magnetic moment of the particle and its easy magnetization axis. The linear dependence of ln(f ) vs 1/T B observed in Fig. 11 indicates that the Neél–Arrhenius model correctly suits the behavior of the M5 sample. From the fitting of the experimental T( f ,H⫽0) data using Eq. 共3兲 and the average particle radii from TEM data, we obtained the values of K eff⫽35.6⫻105 erg/cm3 and ␶ 0 ⫽9⫻10⫺13 s. The resulting effective anisotropy is an order of magnitude larger that the magnetocrystalline anisotropy 共first-order兲 constant of bulk 5 3 magnetite K bulk 1 ⫽1.35⫻10 erg/cm , and this enhancement observed in nanoparticles is customarily associated to surface effects.10,38 However, as will be discussed later, for a spherical particle the contribution of the surface anisotropy should average to zero, and therefore no contribution can be expected from the surface in the present particles. It is worth to notice that, in addition to the contributions from the intrinsic particle anisotropy to E a 共such as shape, magnetocrystalline, or stress anisotropies兲, interparticle interactions 共di-


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polar or exchange兲 can also modify the energy barrier.39 In a previous work on Fe3 O4 particles of d⫽5 nm, diluted in a ferrofluid,40 it was found that dipolar interactions are already noticeable for concentrations of ⬃2 vol % of magnetic particles. When comparing against our data on 具 d 典 ⫽5 nm particles, the larger blocking temperature at low fields observed in our sample further agrees with the presence of strong dipolar interactions due to the higher concentration of magnetic particles. Therefore, the value of K eff obtained for the present nondiluted system clearly contains the effect of particle–particle interactions and must be taken with great caution.

IV. DISCUSSION

Recently, several works on nanometer-sized magnetite particles and films obtained by different techniques have found that the resulting materials may display large differences regarding their magnetic properties. These differences are attributed to changes in structural disorder,41 creation of antiphase boundaries,17 or the existence of a magnetically dead layer at particle surface.42 However, other SQUID and magneto-optical Kerr effect measurements performed on magnetite films43,44 showed that bulk-like behavior can be present for linear dimensions down to 10 nm. The saturation magnetization values found in nanostructured materials are usually smaller than the corresponding bulk phases, provided that no changes in ionic configurations occur. Accordingly, experimental values for M S in magnetite nanoparticles have been reported to span the 30–50 emu/g range, lower than the bulk magnetite value ⬃90 emu/g.45 Many studies have been reported on the origin of the observed reduction in magnetization in fine magnetic particles. The first studies on the decrease in magnetization performed in ␥ -Fe2 O3 by Coey46 showed that this reduction is due to the existence of noncollinear spins at the surface, making the same mechanism appealing for Fe3 O4 . Also, in maghemite fine particles, Morales et al.47 have reported a linear correlation between saturation magnetization and particle size, suggesting that defects at the particle surface can influence the magnetic properties. The present results in Fe3 O4 particles show that M S at 5 K 共where particles are blocked兲 does not scale with d 2 as expected if contributions from a magnetically disordered surface phase were present. Therefore, we also examined the possibility of the observed decrease in M S being due to changes in A- and B-site population, which can change the resulting ferrimagnetic moment. These size effects on the inversion degree of several other spinel systems such as ZnFe2 O4 and CuFe2 O4 are well established48 –50 especially in Zn-ferrite, where large changes in M S and T C have been reported.36,51 Using the notation 关 Fe3⫹ 兴 A关 Fe3⫹ Fe2⫹ 兴 B O4 for the ionic distribution, it can be easily calculated that even for a full inversion of Fe2⫹ ions from B to A sites, a maximum of ⬃20% of change in the net magnetic moment can be expected, which is only half of the ⬃40% observed between the M150 and M5 samples. Thus, this effect does not suffice to explain the observed decrease in M S , and spin disorder at

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the surface and/or spin canting is also needed to explain this reduction. Bulk magnetite is known to be magnetically soft, with values of coercive force H C ⬃200– 400 Oe at room temperature34 that depends on the structural characteristics. For particles below a critical diameter D S a single-domain behavior is expected. For a given temperature, H C decreases from its maximum value H C (D S ) to zero with decreasing particle size due to the superparamagnetic relaxation effects. For the present samples, the onset of SPM relaxation takes place below 具 d 典 ⫽10 nm as inferred from the H C values. At T⫽5 K, the remanence-to-saturation ratio R⫽M R /M S for all samples studied 共see Table I兲 are much smaller than the expected R⫽0.5 value for noninteracting, randomly oriented particles with uniaxial symmetry. Deviations from the noninteracting values are expected, and moreover these low R values reveal the existence of interparticle interactions of antiferromagnetic nature.52 At T⫽5 K all particles are blocked, and the magnetic moments will reverse by rotation. From the field-cooling procedure used in the present measurements, a full alignment of the individual particles should be expected. For Fe3 O4 phase below T V , the magnetocrystalline anisotropy is expected to be uniaxial, and the experimental coercivity field is related to the effective anisotropy constant K eff through the relation H C ⫽2K eff /M S . From the above, valid for noninteracting particles with uniaxial anisotropy, we obtained K eff ⫽3.9⫻104 erg/cm3 . This effective value obtained under the assumption of uniaxial anisotropy is clearly unphysical, since its magnitude is below the first-order magnetocrystal5 3 line anisotropy value K bulk 1 ⫽1.1– 1.3⫻10 erg/cm of bulk 19,53 magnetite. This result implies that the transition from cubic-to-uniaxial crystalline symmetry does not take place even at T⫽5 K, which is also supported by the absence of Verwey transition observed in M (T) curves. We have, therefore, assumed the cubic anisotropy of the material to estimate the particle anisotropy, for which the anisotropy K 1 is related to the effective value by K eff⫽K1/12, provided that K 1 ⬍0.54 For the present particles the above assumption yields K 1 ⫽4.68⫻105 erg/cm3 , which is consistent with K bulk 1 , although increased by the dipolar contribution to the local field. It is interesting to compare these results with the value K eff⫽2.3⫻104 erg/cm3 obtained by Luo et al.40 in Fe3 O4 particles with d⫽5 nm diluted in a frozen fluid, where dipolar interactions are much less important. This value yields K 1 ⫽2.7⫻105 erg/cm3 , closer than our value to K bulk 1 but still larger. We have further checked our results using the same method of Ref. 40, based on the area within the hysteresis cycle, to estimate K eff .55 This method gives K eff ⫽5.2⫻104 erg/cm3 , in reasonable agreement with results from coercive fields. The existence of particle interactions makes the analysis of the single-particle magnetic anisotropy difficult to achieve, since contributions from the neighbor magnetic dipoles to the local field can be even larger than the intrinsic 共crystalline or shape兲 anisotropy. The phenomenological expression generally used for describing the anisotropy K eff of spherical particles of diameter d has been put forward by Bødker et al.,56


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6 K eff⫽K V ⫹ K S , d

共4兲

where K V is the bulk anisotropy energy per unit volume, and K S is the surface density of anisotropy energy. From symmetry arguments, and assuming that surface anisotropy is normal to the particle surface, Bodker et al.56 have demonstrated that a perfect spherical particle should have a zero net contribution from surface anisotropy. This argument was utilized in the same work to explain the value of K S inferred for a system of spherical ␣-Fe nanoparticles, which was six orders of magnitude smaller than the surface anisotropy K S measured for ␣-Fe thin films.56,57 However, Eq. 共4兲 does not take into account the existence of interparticle 共e.g., dipolar兲 interactions, and therefore overestimates K S 共or K eff) in concentrated systems. As shown from TEM pictures, the present Fe3 O4 particles are nearly spherical, so no contribution from the surface should be expected. Consequently, the observed enhancement of the particle anisotropy should be mainly related to the effect of dipolar interactions rather than to a surface with larger anisotropy. The large difference between values of K eff obtained from static and dynamic measurements, 0.39 and 35.6 ⫻105 erg/cm3 respectively, can be explained qualitatively by considering the different magnetic regimes of each measurement. When the applied field is larger than a certain value H 0 that satisfies ␮ H 0 ⫽k B T B , where ␮ is the particle moment and k B is the Boltzmann constant, it can be expected that the magnetic energy dominates the interaction energy. It has been demonstrated40 that for weakly interacting Fe3 O4 particles these two field regimes are separated by H 0 ⬇150 Oe, and for the high field regime the values of T B do not depend on the particle interactions. In the present case, the higher concentration results in larger T B values, and accordingly we calculated H 0 ⬇460 Oe. Therefore, the dynamic value of K eff , obtained at H⬇0, should be largely affected by the presence of dipolar interactions, whereas the static measure is essentially determined in the regime (H ⫽70 kOe) dominated by the external field. Studies on thin-film saturation magnetization at room temperature showed a decrease of the magnetic moment at the interface, which was modeled with nonmagnetic or disordered, ‘‘dead’’ interface layers of 0.7 nm.44 Depth-selective Mo¨ssbauer spectra measurements of 50.0- and 100-nm-thick bulk-like films indicated that the interface, surface, and bulk layers have the same magnetic properties, including the characteristic charge-ordering Verwey transition.58 In a more recent work on ultrathin layers 共thickness ⬃5 nm兲,15 it was also demonstrated that the properties of the surface and interface layers were not different from the interior, i.e., no dead layer existed on these films. Our results point to the latter picture, i.e., the particles have an intrinsic magnetic anisotropy similar to bulk value, although a spin disorder at the surface yields reduction of the magnetic moment. In summary, we have studied the structural and magnetic properties of nearly spherical magnetite particles of different average sizes, showing the gradual evolution from bulk-like to single-domain behavior. Although spin disorder at the surface was needed to explain the reduction of the magnetic

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moment, no evidence was found of enhancement of the magnetic anisotropy at the surface, nor of exchange coupling between particle surface and core. We have found that the static and dynamic properties can be understood by considering changes in the single-particle anisotropy energy E a through the effect of interparticle interactions in these concentrated systems. ACKNOWLEDGMENTS

Three of the authors 共T.S.B., F.C.F., and G.F.G.兲 are thankful to the Brazilian agencies FAPESP and CNPq for providing financial support. 1

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